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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rprmasso | Structured version Visualization version GIF version | ||
| Description: In an integral domain, the associate of a prime is a prime. (Contributed by Thierry Arnoux, 18-May-2025.) |
| Ref | Expression |
|---|---|
| rprmasso.b | ⊢ 𝐵 = (Base‘𝑅) |
| rprmasso.p | ⊢ 𝑃 = (RPrime‘𝑅) |
| rprmasso.d | ⊢ ∥ = (∥r‘𝑅) |
| rprmasso.r | ⊢ (𝜑 → 𝑅 ∈ IDomn) |
| rprmasso.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
| rprmasso.1 | ⊢ (𝜑 → 𝑋 ∥ 𝑌) |
| rprmasso.y | ⊢ (𝜑 → 𝑌 ∥ 𝑋) |
| Ref | Expression |
|---|---|
| rprmasso | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rprmasso.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∥ 𝑌) | |
| 2 | rprmasso.y | . . . 4 ⊢ (𝜑 → 𝑌 ∥ 𝑋) | |
| 3 | rprmasso.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
| 4 | eqid 2737 | . . . . 5 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅) | |
| 5 | rprmasso.d | . . . . 5 ⊢ ∥ = (∥r‘𝑅) | |
| 6 | rprmasso.p | . . . . . 6 ⊢ 𝑃 = (RPrime‘𝑅) | |
| 7 | rprmasso.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ IDomn) | |
| 8 | rprmasso.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
| 9 | 3, 6, 7, 8 | rprmcl 33546 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| 10 | 3, 5 | dvdsrcl 20365 | . . . . . 6 ⊢ (𝑌 ∥ 𝑋 → 𝑌 ∈ 𝐵) |
| 11 | 2, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| 12 | 7 | idomringd 20728 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
| 13 | 3, 4, 5, 9, 11, 12 | rspsnasso 33416 | . . . 4 ⊢ (𝜑 → ((𝑋 ∥ 𝑌 ∧ 𝑌 ∥ 𝑋) ↔ ((RSpan‘𝑅)‘{𝑌}) = ((RSpan‘𝑅)‘{𝑋}))) |
| 14 | 1, 2, 13 | mpbi2and 712 | . . 3 ⊢ (𝜑 → ((RSpan‘𝑅)‘{𝑌}) = ((RSpan‘𝑅)‘{𝑋})) |
| 15 | 7 | idomcringd 20727 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing) |
| 16 | 8, 6 | eleqtrdi 2851 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (RPrime‘𝑅)) |
| 17 | 4, 15, 16 | rsprprmprmidl 33550 | . . 3 ⊢ (𝜑 → ((RSpan‘𝑅)‘{𝑋}) ∈ (PrmIdeal‘𝑅)) |
| 18 | 14, 17 | eqeltrd 2841 | . 2 ⊢ (𝜑 → ((RSpan‘𝑅)‘{𝑌}) ∈ (PrmIdeal‘𝑅)) |
| 19 | eqid 2737 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 20 | 6, 19, 7, 8 | rprmnz 33548 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ (0g‘𝑅)) |
| 21 | 12 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑅)) → 𝑅 ∈ Ring) |
| 22 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑅)) → 𝑋 ∈ 𝐵) |
| 23 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑅)) → 𝑌 = (0g‘𝑅)) | |
| 24 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑅)) → 𝑌 ∥ 𝑋) |
| 25 | 23, 24 | eqbrtrrd 5167 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑅)) → (0g‘𝑅) ∥ 𝑋) |
| 26 | 3, 5, 19 | dvdsr02 20372 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((0g‘𝑅) ∥ 𝑋 ↔ 𝑋 = (0g‘𝑅))) |
| 27 | 26 | biimpa 476 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ (0g‘𝑅) ∥ 𝑋) → 𝑋 = (0g‘𝑅)) |
| 28 | 21, 22, 25, 27 | syl21anc 838 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑅)) → 𝑋 = (0g‘𝑅)) |
| 29 | 20, 28 | mteqand 3033 | . . 3 ⊢ (𝜑 → 𝑌 ≠ (0g‘𝑅)) |
| 30 | 19, 3, 6, 4, 7, 11, 29 | rsprprmprmidlb 33551 | . 2 ⊢ (𝜑 → (𝑌 ∈ 𝑃 ↔ ((RSpan‘𝑅)‘{𝑌}) ∈ (PrmIdeal‘𝑅))) |
| 31 | 18, 30 | mpbird 257 | 1 ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4626 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 0gc0g 17484 Ringcrg 20230 ∥rcdsr 20354 RPrimecrpm 20432 IDomncidom 20693 RSpancrsp 21217 PrmIdealcprmidl 33463 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-rprm 20433 df-subrg 20570 df-idom 20696 df-lmod 20860 df-lss 20930 df-lsp 20970 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-rsp 21219 df-prmidl 33464 |
| This theorem is referenced by: unitmulrprm 33556 |
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